Timeline for Grothendieck on polyhedra over finite fields
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| 2 hours ago | comment | added | user149000 | @mr_e_man I don't understand your definition of $u_0, u_1, u_2$. What exactly do you mean by fundamental domain of the polyhedron, and what are the three planes you are taking the unit normals to to obtain the $u_i$? | |
| Nov 25 at 4:25 | comment | added | mr_e_man | doing similarly for all the basis vectors, we get the reflection matrices $$\sigma_0=\begin{bmatrix}-1&2\cos\tfrac\pi q&0\\0&1&0\\0&0&1\end{bmatrix}$$ $$\sigma_1=\begin{bmatrix}1&0&0\\2\cos\tfrac\pi q&-1&2\cos\tfrac\pi p\\0&0&1\end{bmatrix}$$ $$\sigma_2=\begin{bmatrix}1&0&0\\0&1&0\\0&2\cos\tfrac\pi p&-1\end{bmatrix}$$ and you can combine these to get the rotation matrices $\rho_0,\rho_1,\rho_2$ as in that paper (but of course the basis is different). | |
| Nov 25 at 4:23 | comment | added | mr_e_man | @user149000 - Are you the author of that? In any case, I have a suggestion, a different approach. Let $u_0,u_1,u_2$ be the unit normal vectors to the fundamental domain of the polyhedron $\{q,p\}$ (the polyhedron with $q$-sided polygons, $p$ per vertex). Up to scale, these vectors are dual to $v_0,v_1,v_2$ (vertex, edge, face). The dot products are $u_0\cdot u_1=-\cos\tfrac\pi q$, $u_1\cdot u_2=-\cos\tfrac\pi p$, $u_0\cdot u_2=0$. Note, the dihedral angle does not appear. The reflection of $u_1$ along $u_0$ is given by $$\sigma_0(u_1)=u_1-2(u_1\cdot u_0)u_0=u_1+(2\cos\tfrac\pi q)u_0;$$ | |
| Nov 22 at 4:48 | comment | added | user149000 | @mr_e_man See arxiv.org/abs/2304.03345 | |
| Nov 20 at 20:00 | comment | added | mr_e_man | @user149000 "and each generator is simply a linear polynomial in the flag" - Can you clarify? | |
| May 29, 2022 at 20:04 | history | edited | LSpice | CC BY-SA 4.0 |
Inlining Grothendieck quotes
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| May 29, 2022 at 18:18 | answer | added | user234212323 | timeline score: 3 | |
| Jul 14, 2021 at 22:25 | comment | added | user149000 | He defines maps in a previous section essentially as graphs on surfaces. As he explains in the same section, one can view a regular polyhedron as defined simply by its (oriented) automorphism group acting on a flag (vertex + center of side + center of face), and each generator is simply a linear polynomial in the flag. The polyhedron chosen determines the angles, which determines the polynomial. One can consider these same equations in various characteristics, which gives you the specializations. However I don't know what description he refers to (maybe you need a projective context). | |
| Feb 7, 2015 at 15:27 | comment | added | Matthias Wendt | Maybe the following paper is relevant: B. Monson, E. Schulte. Reflection groups and polytopes over finite fields I. Advances Applied Mathematics 33 (2004), 290--317 | |
| Feb 7, 2015 at 15:23 | comment | added | Matthias Wendt | Could the singular characteristic have something to do with the modular representation theory of the automorphism groups? Also, isn't it more natural to relate the 27 lines to $E_6$ instead of $E_7$? | |
| Feb 7, 2015 at 15:03 | history | asked | tghyde | CC BY-SA 3.0 |